1 Descriptive Statistics

1.1 By Month

1.2 Distance

1.3 Time

1.4 Velocity

1.5 Angles

2 Correlated random walk

Process Model

\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]

2.1 Parameters

For each individual:

\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]

For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]

2.2 Behavioral States

\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]

\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]

\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix} \]

2.3 Environment

Behavioral states are a function of local environmental conditions. The first environmental condition is ocean depth. I then build a function for preferential foraging in shallow waters.

It generally follows the form, conditional on behavior at t -1:

\[Behavior_t \sim Multinomial([\phi_{traveling},\phi_{foraging}])\]

With the probability of switching states:

\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}} + \beta_{Month,1} * Ocean_{y[t,]} + \beta_{Month,2} * Coast_{y[t,]}\]

\[logit(\phi_{foraging}) = \alpha_{Behavior_{t-1}} \]

Following Bestley in preferring to describe the switch into feeding, but no estimating the resumption of traveling.

The effect of the environment is temporally variable such that

\[ \beta_{Month,2} \sim ~ Normal(\beta_{\mu},\beta_\tau)\]

2.4 Continious tracks

The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.

How did the filter change the extent of tracks?

sink(“Bayesian/Multi_RW.jags”) cat(" model{

#Constants
pi <- 3.141592653589

##argos observation error##
argos_prec[1:2,1:2] <- inverse(argos_sigma*argos_cov[,])

#Constructing the covariance matrix
argos_cov[1,1] <- 1
argos_cov[1,2] <- sqrt(argos_alpha) * rho
argos_cov[2,1] <- sqrt(argos_alpha) * rho
argos_cov[2,2] <- argos_alpha

for(i in 1:ind){
for(g in 1:tracks[i]){

## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec)

#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)

###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs

#Process Model for movement
for(t in 2:(steps[i,g]-1)){

#Behavioral State at time T
logit(phi[i,g,t,1]) <- alpha_mu[state[i,g,t-1]] + beta[Month[i,g,t-1],state[i,g,t-1]] * ocean[i,g,t] + beta2[Month[i,g,t-1],state[i,g,t-1]] * coast[i,g,t]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])

#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])

#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])

#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}

#Final behavior state
logit(phi[i,g,steps[i,g],1]) <- alpha_mu[state[i,g,steps[i,g]-1]] + beta[Month[i,g,steps[i,g]-1],state[i,g,steps[i,g]-1]] * ocean[i,g,steps[i,g]] + beta2[Month[i,g,steps[i,g]-1],state[i,g,steps[i,g]-1]] * coast[i,g,steps[i,g]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])

##  Measurement equation - irregular observations
# loops over regular time intervals (t)    

for(t in 2:steps[i,g]){

# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){ 
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]

#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec)
}
}
}
}
###Priors###

#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)

##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)

# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi

# prior for theta in 'foraging state'    
theta[2] <- (tmp[2] * pi * 2)

##Move persistance
# prior for gamma (autocorrelation parameter) in state 1
gamma[2] ~ dbeta(1.5, 5)        ## gamma for state 2
dev ~ dbeta(1,1)            ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[1] <- gamma[2] + dev      ## gamma for state 1


#Monthly Covaraites
for(x in 1:Months){
beta[x,1]~dnorm(beta_mu[1],beta_tau[1])
beta[x,2]<-0
beta2[x,1]~dnorm(beta2_mu[1],beta2_tau[1])
beta2[x,2]<-0
}

##Behavioral States

#Hierarchical structure across motnhs
#Intercepts
alpha_mu[1] ~ dnorm(0,0.386)
alpha_mu[2] ~ dnorm(0,0.386)

#Variance
alpha_tau[1] ~ dt(0,1,1)I(0,)
alpha_tau[2] ~ dt(0,1,1)I(0,)

#Slopes
## Ocean Depth
beta_mu[1] ~ dnorm(0,0.386)
beta_mu[2] = 0

# Distance coast
beta2_mu[1] ~ dnorm(0,0.386)
beta2_mu[2] = 0

#Monthly Variance
#Ocean
beta_tau[1] ~ dt(0,1,1)I(0,)
beta_tau[2] = 0

#Coast
beta2_tau[1] ~ dt(0,1,1)I(0,)
beta2_tau[2]  = 0


#Probability of behavior switching 
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]

##Argos priors##
#longitudinal argos error
argos_sigma ~ dunif(0,10)

#latitidunal argos error
argos_alpha~dunif(0,10)

#correlation in argos error
rho ~ dunif(-1, 1)


}"
,fill=TRUE)

sink()

##      user    system   elapsed 
##  1328.321     6.119 58037.474

2.5 Chains

2.5.1 Compare to priors

2.6 Parameter Summary

##    parameter         par         mean         lower        upper
## 1   alpha_mu alpha_mu[1] -0.417332989 -0.8344519592  0.030073584
## 2   alpha_mu alpha_mu[2] -1.559019914 -1.9031016985 -1.226445370
## 3       beta   beta[1,1]  0.163465079 -0.9410132901  1.240013940
## 4       beta   beta[2,1] -0.152877871 -1.0596105189  0.589875478
## 5       beta   beta[3,1]  0.424846885 -0.1799641752  1.207613452
## 6       beta   beta[4,1]  0.309580913 -0.3686511856  1.132224573
## 7       beta   beta[5,1]  0.378093067 -0.2160579311  1.090667355
## 8       beta   beta[6,1]  0.159160398 -1.1964223866  1.525005390
## 9       beta   beta[1,2]  0.000000000  0.0000000000  0.000000000
## 10      beta   beta[2,2]  0.000000000  0.0000000000  0.000000000
## 11      beta   beta[3,2]  0.000000000  0.0000000000  0.000000000
## 12      beta   beta[4,2]  0.000000000  0.0000000000  0.000000000
## 13      beta   beta[5,2]  0.000000000  0.0000000000  0.000000000
## 14      beta   beta[6,2]  0.000000000  0.0000000000  0.000000000
## 15     beta2  beta2[1,1]  0.015018291  0.0037168885  0.029208777
## 16     beta2  beta2[2,1]  0.012478169  0.0024904166  0.022855093
## 17     beta2  beta2[3,1]  0.007715217 -0.0044199308  0.018900719
## 18     beta2  beta2[4,1]  0.012174000  0.0014702708  0.023618058
## 19     beta2  beta2[5,1]  0.011151502  0.0011736272  0.022313245
## 20     beta2  beta2[6,1]  0.011840779 -0.0111648448  0.042159874
## 21     beta2  beta2[1,2]  0.000000000  0.0000000000  0.000000000
## 22     beta2  beta2[2,2]  0.000000000  0.0000000000  0.000000000
## 23     beta2  beta2[3,2]  0.000000000  0.0000000000  0.000000000
## 24     beta2  beta2[4,2]  0.000000000  0.0000000000  0.000000000
## 25     beta2  beta2[5,2]  0.000000000  0.0000000000  0.000000000
## 26     beta2  beta2[6,2]  0.000000000  0.0000000000  0.000000000
## 27  beta2_mu beta2_mu[1]  0.012014160 -0.0001008535  0.024494507
## 28  beta2_mu beta2_mu[2]  0.000000000  0.0000000000  0.000000000
## 29   beta_mu  beta_mu[1]  0.203896790 -0.4317957328  0.863227270
## 30   beta_mu  beta_mu[2]  0.000000000  0.0000000000  0.000000000
## 31     gamma    gamma[1]  0.939972488  0.9023347576  0.981458840
## 32     gamma    gamma[2]  0.151425865  0.1035755270  0.190875651
## 33     theta    theta[1] -0.021808333 -0.0449621893  0.001247219
## 34     theta    theta[2]  0.323726975  0.1960557078  0.497285833

3 Behavior and environment

3.1 Hierarchical

3.1.1 Ocean Depth

3.1.2 Distance to Coast

3.1.3 Interaction

3.2 By Month

3.2.1 Depth

Just the probability of feeding when traveling.

Just mean estimate.

3.2.2 Coast

Zooming in on the top right plot.

Just mean estimate.

4 Behavioral Prediction

4.0.1 Correlation in posterior switching and state

4.1 Spatial Prediction

4.1.1 Per Animal

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4.2 Log Odds of Foraging

4.2.1 Ocean Depth

4.2.2 Distance From Coast

4.3 Autocorrelation in behavior

4.4 Behavioral description

4.5 Predicted behavior duration

4.6 Location of Behavior

5 Environmental Prediction - Probability of Foraging across time

5.1 Bathymetry

5.2 Distance to coast

5.3 All variables

6 Overlap with Krill Fishery

6.1 By Month

6.2 Change in foraging areas

Jan verus May

Red = Better Foraging in Jan Blue = Better Foraging in May

6.2.1 Variance in monthly suitability

6.2.2 Mean suitability

6.3 Monthly Overlap with Krill Fishery

##             used   (Mb) gc trigger    (Mb)   max used    (Mb)
## Ncells   1967099  105.1   16642152   888.8   40630256  2169.9
## Vcells 517438892 3947.8 1443816256 11015.5 1575715162 12021.8